Production of Robots by means of Robots.

(Sorry about the title. The devil made me write it.)

What are we afraid of? Let's think about the worst-case, nightmare scenario for the distribution of income.

Assume that all capital is robots, and robots are perfect substitutes for human workers. One robot can produce everything and anything one human worker can produce. And that includes producing more robots.

And assume that every year the technology of robot production improves, so that it takes less and less time for one robot to produce another robot.

That sounds nightmarish, right? Because robots will get cheaper and cheaper, and drive down human wages?

Well, no. They won't. Or rather, it all depends. It depends on whether we add other forms of capital, or land, to the model.

Labour and Robots only.

Let's start out by ignoring land. And the only form of capital is robots. You can produce everything with just human or robot labour.

The production function is: C + I/a = L + K and Kdot = I

where C is consumer goods produced per year, I is robots produced per year, a is a parameter which increases over time as technology improves and robots get easier to produce, K is the stock of robots, and L is the number of human workers.

There's another way to look at the parameter a. It's the rate at which robots can reproduce themselves if they aren't producing consumption goods instead. (I'm assuming that robots can't, er, reproduce and make chewing gum at the same time.)

Let's measure wages in terms of consumption goods. Because consumption is what people care about. Robots and humans earn the same wages. Since both robots and human workers produce one unit of consumption goods per year (or per day, or per hour, or whatever) their Marginal Products and wages will be one unit of consumption goods too.

W = 1

In this simple model, improving technology for producing robots has no effect whatsoever on wages.

Not at all nightmarish, is it?

It will however have an effect on the rate of interest.

We know that the price of a robot in terms of consumption goods will be 1/a. (That's because I have assumed a linear PPF between consumption and robots, so the opportunity cost of producing one extra robot is always (1/a) units of consumption).

Suppose a is rising over time, so (1/a) is falling at rate g. We know that each robot earns 1 unit of wages per year. So the rate of interest (measured in terms of consumption goods) Rc must equal the rate of return on owning a robot, which is annual robot wages (1), divided by the price of a robot (1/a), minus the rate of capital losses from the falling price of robots, so:

Rc = a - g.

In this simple model, the rate of interest is determined by the rate at which robots reproduce, and by the rate of change of the rate at which robots reproduce. The bigger is a (the quicker robots reproduce) the higher the rate of interest. The faster a is rising (the quicker the rate of technological change in robot reproduction) the lower the rate of interest. If g is positive but constant, the rate of interest will be rising over time.

It's simpler if we measure interest rates in terms of robots, Rr, because then we can ignore the fact that the price of robots will be falling over time. Since one robot can produce a robots per year,

Rr = a

The interest rate, measured in terms of robots, will be rising over time if technological change increases the rate at which robots reproduce.

Labour and Robots plus other Capital.

Robots are a form of capital goods that are perfect substitutes for labour. What happens if we introduce a different form of capital that is a complement to labour?

The simplest way to do this is to assume there is a one-year lag between humans and robots doing the work and the extra consumption and new robots being produced. So the production function now becomes:

C(t) + I(t)/a = L(t-1) + K(t-1)

The wage, measured in terms of current consumption, now becomes the present value of the (future) Marginal Product of Labour:

W = 1/(!+Rc)

The rate of interest Rc must equal the rate of return on owning a robot, which is the wage of a robot (W) divided by the price of a robot (1/a), minus the capital losses from the falling price of robots, g:

Rc = a/(1+Rc) - g

I think (somebody please check my math) that Rc, as before, is increasing in a and decreasing in g. That means that if a is growing at a constant rate, the rate of interest will be rising over time.

And, since W=1/(1+Rc), that means that wages (in terms of consumption) will be falling over time.

OK. That's a much more nightmarish scenario. For those who only own their own labour.

But it's not very realistic, for recent years, because real interest rates (deflated by the CPI) have not been rising. They have been falling.

Labour and Robots plus Land.

OK, let's scrap the lag in the production function, but put land (Natural Resources, N), along with labour plus robots, into a Cobb-Douglas production function:

C + I/a = (L + K)b.N1-b

It's a constant returns to scale production function, but holding land fixed we get diminishing marginal returns to labour plus robots. (I have implicitly assumed, by making the PPF between C and I linear, that producing consumption goods and robots are equally land-intensive.)

The human (or robot) now earns a wage equal to the Marginal Product of Labour:

W = b(N/(K+L))1-b [edited to fix math error spotted by Kathleen.]

As the number of robots increases, the wage gets driven down by diminishing returns, just as in Malthus/Ricardo, except it is the robot population that is increasing over time, if people save and invest in building more robots.

With a little bit of math, we can show that human plus robot workers earn a constant share b of total output, and landlords earn the remaining constant share (1-b). But as more robots are built, and the robot/human ratio K/L rises, human workers earn a decreasing share of b. And as total output expands, land rents per acre rise.

And the rate of interest is:

Rc = ab(N/(K+L))1-b - g [edited to fix math error]

To figure out whether Rc is rising or falling over time we need to figure out if the growing stock of robots is making the denominator grow more or less quickly than the numerator of the first term. And that will depend on people's consumption/savings choice, which in turn depends on their intertemporal consumption preferences. The math is beyond me, but I'm pretty sure the effect could go either way. (To figure it out, we need an additional equation representing intertemporal preferences in which Rc is an increasing function of the growth rate of consumption.)

Anyway, if you are looking for a nightmare scenario that is at least vaguely realistic, robots alone won't do it. I think you need to go back to Malthus/Ricardo, and put land back into the model.

That's what I was trying to say way back in this old post. I've just said the same thing with more math.

Comments

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Nick,

Instead of these mathematical models where it's always hard to tell when the underlying critical assumptions diverge from reality, why don't we try an elementary analysis of a hypothetical world without land rent?

Suppose we invent a technology that enables us to open portals towards an infinite number of uninhabited parallel Earths in other dimensions, at negligible cost. Now if you want oil, you just need to pay the cost of the drilling machines and labor, teleport them for free into the empty Arabian desert of a parallel Earth, and oil will be flowing through the portal. Or, if you want a house, you just pay the contractors to go to a parallel world with the materials and build it, and afterwards you commute through the portal from home to wherever you want to go.

Imagine now this no-land-rent economy at a technological level otherwise similar to the early 21st century developed world. What would be the possible consequences of the robot scenario?

Nick, back to your point that "oil is land, too": the math gets quite a bit more complicated once you start trying to factor in greenhouse gas emissions and climate change scenarios. Also, you're not going to be able to use your Cobb-Douglas if you actually want to consider the interdependency of the inputs (Costanza, Ayres and Kneese) rather than pretending they are independent, which of course they are not.

Before indulging in all that complicated math, though, it might be worthwhile contemplating that the machinery question and the fuel efficiency/rebound paradox are two sides of the same coin. Malthus was "wrong" about peak population because of peak coal and Jevons was wrong about coal because of petroleum. Maybe Hubbert will be wrong about peak oil because of climate change -- which will make Malthus right about population after all!

2. The real world is partly like that model, if technical change is "land-augmenting", which to some extent it is. Stick another technology parameter in front of N, and suppose it increases over time. That's how we have driven back the Malthusian spectre. So far!

Here's the proposition, the creation of wage-labor "jobs" is coupled with the consumption of hydrocarbon-based fuel. This is an historical-empirical fact that anyone can confirm for themselves by looking at the data for the last 200 odd years. It is neither rocket science nor brain surgery.

Although it may be theoretically possible --"in principle" -- to uncouple or decouple job creation from fuel consumption, no one has done it or shown how it could be done. It is science fiction. But economists persist in assuming that it will happen "automatically," so to speak, through market-generated technological substitution. Or, in plain language, someone will invent a perpetual motion machine. That isn't even theoretically possible!

The Robots symbolize the age-old anxiety about machinery displacing human labor, throwing people out of their jobs and thus depriving them of their livelihood.

The Rift refers to the metabolic rift between town and country, between industrial processes and natural rhythms, biomes and landscapes. Even a good thing can become too much of a good thing toxic in the wrong place at the wrong time.

The Rebound is a descriptive synonym for what is otherwise known as the Jevons paradox, the idea that increases in the fuel efficiency of machinery will paradoxically lead to increased consumption of the fuel because it will make the fuel, in effect, cheaper. And, as Dorning Rasbotham proclaimed 232 years ago, "A cheap market will always be full of customers."

The robot and the rebound are tied to each other continuously in a kind of metabolic Möbius strip. On "one side" are fuel efficiency and consumption and on the other are labor displacement (by machinery) and re-absorption (through market extension). But an ant -- or an unemployed worker -- crawling along this strip would eventually return to its starting point only after traversing both sides.

Nick, the most nightmarish scenario is the one where cost of reproduction of robots is lower then the subsistence wage. It takes many years of feeding and sheltering and training until your average human becomes productive, and then there is more of the same. If robots could do exactly same things for lower costs than is the level of human subsistence, then we end up in a potentially nightmarish scenario. People will become horses of this dystopia - too expensive and too much of a hassle to bother using for a real work.

But what if robots are not just perfect substitutes for humans, but increasingly superior to humans, not just in volume of output and productivity, but in quality - in a non-substitutable way?

Say like car makers don't let human welders near their cars, because robots are welding so much better. Or Apple won't allow humans to assemble the iPhone 6, because machines are orders of magnitude more precise.

If that happens - and every sign is telling us that it's happening on an increasing scale - then the friendly co-existence of robots and humans in your equations ends quickly and possibly exponentially.

In that case human labor will not compete against a rising robot population size, but will compete for a rapidly shrinking pool of jobs that robots cannot do yet or which humans can still do.

The "nightmare scenario" is then for most of humanity to be in that large group - with wages eventually driven below subsistence.

JV: But a falling *price* of robots does not necessarily mean a falling *wage* for robots (or humans). Look at my first model, where the cost of reproducing robots keeps on falling but has zero effect on the wage. Instead it causes the rate of interest to rise.

Put differently, the "nightmare scenario" is not humans competing against robots and the rent of land.

The nightmare scenario is a growing human population competing not against robots, but against other humans, for the rapidly shrinking pool of jobs that robots have not monopolized via their superior work quality yet.

In other words, the worst effect of robots is not price, but the (relative) erosion of the human skill set that can be used by humans to get a job - any job.

Anon: until you buil;d a model showing that, I still say you are wrong. Suppose someone owns a robot that can produce everything they need and want. OK, they buy nothing from the rest of us, but they can't sell anything to the rest of us either, because we can't afford to buy it. So waht? That person is better off, but the rest of us just keep on trading among ourselves, doing as well as we were before.

That's why you need to build a model. Because until I built that model above, showing people were wrong, everyone (I conjecture) was saying "If robots get cheaper to produce, that means the price of robots will fall (correct), and that means wages will fall (incorrect).

C_h is consumer goods produced by human workers. L is human workers. "r" is the "robot quality factor", which increases in time as robots improve."

Stop right there. You have just assumed that the new technology makes human workers less productive than before. If that were the case, why wouldn't human workers say "Stuff this, let's go back to the old technology, and let the robots do what they want!"?

Nick: Your model is correct that falling robot prices don't mean that wages will necessarily fall. But what actually happens to wages depends on exogenous variables of investment and the rate of interest, whose adjustment is not "automatic." See Marx: "The Theory of Compensation as Regards the Workpeople Displaced by Machinery" and Keynes, "Is the Economic System Self-Adjusting?"

Nick: Sorry my bad. I was already assuming land in the model. If we have infinite space to expand and there are only robots and humans and everybody capable of human labor can create things from nothingness with ever increasing productivity, then obviously we will get richer and richer.

Add land or some other rent-like production factor (holders of patents etc.) and you will end up with owners of this production factor owning everything and everybody will be else driven to starvation because their wage will be vastly lower then their productivity - with surplus being captured by our rentier.

"Anyway, if you are looking for a nightmare scenario that is at least vaguely realistic, robots alone won't do it. I think you need to go back to Malthus/Ricardo, and put land back into the model."

But if you're looking to avoid the nightmare scenario, you only need to go back to Marx and "wake up from the dream."

"Our programme must be: the reform of consciousness not through dogmas but by analyzing mystical consciousness obscure to itself, whether it appear in religious or political [or Cobb-Douglas production function] form. It will then become plain that the world has long since dreamed of something of which it needs only to become conscious for it to possess it in reality. It will then become plain that our task is not to draw a sharp mental line between past and future, but to complete the thought of the past. Lastly, it will becomes plain that mankind will not begin any new work, but will consciously bring about the completion of its old work."

In reality robots would produce significantly more than humans would, although the prices for these products might only fall so far because of price floors on the underlying natural resources due to limits. I don't see how your model says wages would be unaffected, seems like as people are simply put into unemployment human wages to go zero in the sectors that the robots produce in.

But then why are any humans going to be working any way? Suppose you're a commercial airline and have access to robots that replace pilots. You don't hire any pilots and you just have robots fly all your planes. You still have to pay for fuel etc. and so there are some savings to pass onto customers but the price of a ticket doesn't fall to anywhere close to zero no matter how cheap robots get. The pilots are totally replaced though so their wages go to zero.

Now, this is happened in industries before as technology has made certain professions obsolete. But of course if the robots could do just about any job this effect would occur across all sorts of sectors. Perhaps wages of still working people wouldn't change but significantly fewer people would be working.

The comments seem to have missed it out, but I think Rowe is trying to link it back to the argument invoking real interest rates. We can all make up plausible stories regarding income inequality and Farewell-to-Alms-style horse obsolescence.

Anyway. In the land model, surely the real interest rate must track the rate at which the real value of land rises? Otherwise you would borrow and buy land, or vice versa.

[Stop right there. You have just assumed that the new technology makes human workers less productive than before. If that were the case, why wouldn't human workers say "Stuff this, let's go back to the old technology, and let the robots do what they want!"?]

It happens, and the simple answer is "because the workers are just workers, not management, and not owners." And by the way, the new technology does on occasion make human workers less productive - measured separately from that technology, which economists don't seem to have thought to do heretofore - but it certainly makes them less _valuable_, since it provides a cheaper alternative to that labor, where it's an alternative at all. And if an aggregate production function is improved, even while the human worker becomes less productive, the new technology is at least economically justified. Morally or ethically, no, but economically, yes. At least to those who make this call. And, surprise, those people are not often economically literate; they do things that don't make sense. But they do like gadgets - new technology is nothing if not gadgets. They like new technology like first graders like shiny new toys.

I build models, too. They're called "software." And they're readily tested, unlike yours. One of those is the Apache web server that your blog is running on:

It seems to me you're hypothesizing about how things might be - but it seems you're doing it in an effort to discount how things actually are. That alone makes your model not just wrong, not just worthless, but damaging. If you want to build a model, build one that can actually be descriptive of the real world - really descriptive of it, not just limited to your admittedly meager math skills.

My models do real things, like run your blog. Yours, at this level, are like first graders drawing with crayons. So you can complain all you want, "my model says you are wrong," but all that complaint does, in my opinion, is make you sound like a first-grader.

It really seems to me you haven't tried very hard to look around you and see what this new technology actually does - even just for you. And if you haven't looked, you haven't measured, and if you haven't measured, your model can't possibly be realistic.

The real economic issue isn't robots - which, by the way, are now even driving cars and flying planes without human occupants. The real economic issue is that there's no such thing as a workplace without a computer in it, pretty much anywhere. When I first started programming, folks were talking about the reasonable claim, at the time, that there would never be a need for more than a dozen computers in the world. I have more than that myself. And just about any literate person in the US these days has or uses at least one.

Your model reminds me of that "no more than a dozen computers" claim.

It doesn't take much of an impact to explain unemployment rising from, say, 4% to 10%. But that much of an impact - from whatever combination of causes - is apparently not one that should be ignored. And you're doing nothing if not trying to ignore one of those factors.

And John provides a demonstration of why growth theory is macro: because the argument isn't about microscopic labour-market interactions at all. The argument is about the sign of the real interest rate!

In your model, the production of robots increases exponentially. The result in the limit is that consumption goods relative to all goods and human labor relative to all labor both go to measure 0. At that point you can make no prediction about them, since they are statistical anomalies. Of course this is a ridiculous outcome, but to prevent it, production of consumer goods has to increase and their price has to decline, or their production has to consume more labor. In any case the human share of total labor marginal product goes to 0. If labor is paid according to marginal product, then the human share of labor pay goes to 0, even if it remains constant in absolute terms. At some point human labor will of too little use to bother with. Robots are much easier to manage than humans.

Labor will also be worse off if the cost of living increases relative to marginal product, but it's hard to make a prediction about this.

With the portals-to-infinite-land world, I was aiming for a situation of the second kind in your list, i.e. one where capital is important, but not land. I've been trying to map your scenario of rising interest rates and falling wages to this world, but the more I think about it, the less sense it makes. Take for example robots whose reproduction is very labor-intensive so that "a" is only slightly above 1, and that it's roughly constant, so that "g" is close to zero. Your model then implies interest rates given approximately by the golden ratio. But why?

Only a miro-economist would try to prove that nobody needs to worry about robots driving down wages. Then admit as an aside that if land exists the robots can drive wages below subsistence level. You may live in a landless world. The rest of us don't.

are those who foresee a nightmare scenario envisaging the robots are owned by a small minority of people?

if we do go down that road, I wouldn't be surprised if there has to be some sort of revolution to redistribute the fruits of our amazing technological production sector. After all, we have to get from here, to here: http://en.wikipedia.org/wiki/The_Culture

what if the robots are really small and require very little land and are perfectly happy in the middle of Mongolia, or maybe everything is built on floating platforms in the ocean or sky, and raw material are mined from asteroids.

david hits the nail on the head. I'm just going to edit his comment slightly, to get:

"And John [and many commenters] provides a demonstration of why growth theory is [general equilibrium theory]: because the argument isn't about [partial equilibrium] labour-market interactions at all."

You can't use partial equilibrium reasoning to solve a general equilibrium question. Classic fallacy of composition. "If robots can do my job the demand for my labour falls, and my wages fall, therefore if robots can do everybody's job the demand for everybody's labour will fall and everybody's wages will fall!"

"If I sit down in the theatre I will see worse, therefore if everybody sits down everybody will see worse".

If robots can do my job, my wages fall, but the price of the goods I am producing falls too (relative to the general price level). That raises everybody else's real wages.

"Assume that all capital is robots, and robots are perfect substitutes for human workers. One robot can produce everything and anything one human worker can produce. And that includes producing more robots.

And assume that every year the technology of robot production improves, so that it takes less and less time for one robot to produce another robot.

That sounds nightmarish, right? Because robots will get cheaper and cheaper, and drive down human wages?"

The bit I don't understand. If the robots are building the robots that build the robots and doing so ever more cheaply. And it becomes robots all the way down. The the price of consumer goods becomes spit. Because no one has to do any labour in order for there to be a cornucopian world. Everything's made by the machines at the extreme. True communism has arrived in fact.

Another way of putting this is to look at wages in terms of consumption. If the result of the robot farm, mill, baker and truck delivery to the house is a loaf of bread at 1 cent then who the hell cares that wages to humans have fallen by 98%?

And the only way that entirely robotic production isn't going to cause a collapse in prices is if "capital", those owners of those robots, can collude to stop it.

We're back with Smith and "businessmen seldom" etc.

Further, we've actually seen this happen before. With farming. Once that mechanised then shouldn't all of the money have gone to the landowners? They just have to use machines after all: exactly analogous to "capital" with the robots. We should have ended up with the landowners having all the money and the rest of us none.

But we didn't. We actually have to subsidise the farmers now as competition using the new technology drove food prices down. Mechanisation of farming produced the greatest rise in the living standards of the average man ever.

I'm simply not getting why anyone thinks that the mechanisation of manufacturing (or of services, whatever) won't produce the same result.

Peter N: "Both C and L are functions of t so the model as it stands is underdetermined. Obviously C matters. Constant C and C growing proportional to K are rather different."

That is partly true. Given K(t) and L(t), all my 3 models are fully determined at time t, except for the mix of output between C(t) and I(t). We need to add some sort of savings function (intertemporal preferences) to determine the C/I mix, and how the economy will evolve over time.

Note that I had to assume a linear PPF between I and C to determine the price of robots and the rate of interest. If I had instead assumed a more general non-linear PPF I would have needed an extra equation for the savings function to determine those two variables (and also to determine W, in the second model).

Tim: my guess is that you are implicitly assuming a representative agent model. Which is OK, but may leave some stuff out. The representative agent would have growing C, and stagnant of declining W, so if leisure is a normal good (it seems to be) the income and substitution effects combined will lead to labour supply falling towards zero.

If you instead do classical class analysis (workers, capitalists, landlords), treating the functional distribution of income as equivalent to the personal distribution of income, you would get a different perspective.

The truth is somewhere in between, of course. Like me: I belong to all 3 classes, until I retire.

"The bit I don't understand. If the robots are building the robots that build the robots and doing so ever more cheaply. And it becomes robots all the way down. The the price of consumer goods becomes spit."

What's your numeraire? The price of consumption goods in terms of what? Note that I am taking consumption goods as the numeraire.

Luis: "are those who foresee a nightmare scenario envisaging the robots are owned by a small minority of people?"

Yes. I think so. And maybe many implicitly assume that the functional distribution of income is the same as the personal distribution of income. They have a class analysis that would apply to 18th/19th century England.

If production by robots was less land-intensive than production by human labour, I think that would change my third model a lot. I haven't done the math though.

I can put my parameter "a" underneath I or in front of K. It doesn't make any difference, except for vintage effects. My way of doing it handles the vintage effects better. Old robots don't get better over time. But new robots are better than old robots. So if two new robots are twice as productive as one old robot, but cost the same to produce, it's easier to redefine robots so that one new robot is really two robots, each costing half as much to produce.

Vladimir: If humans have a comparative advantage over robots in producing robots, and robots have a comparative advantage over humans in producing consumption goods, then:

1. Robots aren't identical to human workers (I assumed they are).

2. The PPF between C and I will be convex, and so you won't be able to figure out the price of robots in terms of consumption goods (and the rate of interest) without knowing the C/I ratio, which will depend on savings preferences.

(BTW, I was re-reading your old comments on my old post that I linked to above. They were good comments.)

david: "Anyway. In the land model, surely the real interest rate must track the rate at which the real value of land rises? Otherwise you would borrow and buy land, or vice versa."

That is true, but you have causality reversed (in this model). We know that land rents will be rising over time as the stock of robots+human workers increases over time. And it is true that the rate of return on owning land must equal the rate of return on owning robots, and both must equal the rate of interest. But whereas the price of robots is pinned down by the linear slope of the PPF between C and I (in this simple model), there is no technology for producing more land, so the price of land is not pinned down by technology. The rate of interest determines the price of land, not vice versa.

(For a very different model, see my old post on Dutch Capital Theory, where the price of land is pinned down by intertemporal preferences (or jointly determined by preferences and the curvature of the PPF for producing new land).

BTW, speaking of Dutch capital Theory, where are the English Capital Theorists? I would have thought they would be all over the comments by now, given that title? Maybe they are plotting strategy for an attack en masse.

Please excuse John. He grew up watching big guys push kilobytes through the intertubes and earn good wages. Now he's seeing robots push gigabytes for less. You can't placate him by pointing out that bandwidth is cheap for everyone.

Nick, have you noticed an uptick in traffic recently? Your thermostat post was posted on Hacker News [http://news.ycombinator.com/item?id=4915751] a few days ago. The comments on your post are closed. I think the ycombinator ones are still open. Enjoy!

Daphne: "Only a miro-economist would try to prove that nobody needs to worry about robots driving down wages. Then admit as an aside that if land exists the robots can drive wages below subsistence level. You may live in a landless world. The rest of us don't."

jeeez! Talk about missing the whole point of this post! That bit about land wasn't an aside. That was the main point of the post! Plus, you accuse *me* of living in a landless world? I'm the one that keeps yammering on about land! (And how many acres do you own?)

The long run scary scenario is that robots replace workers. Presumably this is scary because workers work to earn an income, if robots replace them workers no longer have an income. How is this not a sort of impossible long-run outcome? Who exactly will buy the output of the robots (presumably the output of these robots is owned by some small minority of capital owners)?

Why isn't the impossibility of this outcome a partial analogy for what is happening now and the reason inequality hurts growth. Capital earned increasing share of national income recently, it seems to me this has hit something of a limit, not just politically, but as a practical economic threshold where now in order to increase its share of the pie, the size of the pie is shrinking as demand from workers reduced incomes shrink.

The short run robot scenario is possibly more scary than the long run (if indeed the long-run scenario is impossible) for all the reasons the past 30 years have been scary for the middle class.

"The Rebound is a descriptive synonym for what is otherwise known as the Jevons paradox, the idea that increases in the fuel efficiency of machinery will paradoxically lead to increased consumption of the fuel because it will make the fuel, in effect, cheaper."

The Jevons paradox, as far as I know, describes a rebound of >100%, which is also called "backfire" in the literature.

"the 'rebound effect' is just another name for 'demand curves slope down'."

That's a little misleading, I think. The rebound describes the reaction of fuel demand with respect to changes in fuel efficiency. In a diagram with fuel on the x-axis and Marginal Benefits of fuel use (=demand) on the y-axis, the curve should be downward-sloping; whether there is backfire or not depends on whether an increase in efficiency will move the curve to the North-East (backfire) or to the South-West (no backfire).

If robots can do my job, my wages fall, but the price of the goods I am producing falls too (relative to the general price level). That raises everybody else's real wages.

Right but that's the catch isn't it? No one who is concerned about automation taking jobs away is concerned about some overall effect. Of course some people will remain employed and see their real wages increase, but the "nightmare" scenario has always been a distributional issue. That's not some secondary issue that economists always like to wave away that's the heart of the matter. It doesn't help your real wages if you have 0 income. What happens in the long run equilibrium is actually not relevant because people's lives are destroyed today.

The "fallacy of composition" is not really that much of a fallacy either. You can say that the lower price of goods raises real income which means people who continue to receive income will have more of it to spend. However, the robots still fulfil all demands. It's not really hard to see that the owners of robots and maybe some robot-related professionals will see their real incomes rise but that doesn't debunk the nightmare scenario.

Tim Worstall: avoiding collapse was contingent. By luck and pluck We put in place policies to avoid that: getting rid of the Corn LAw thus tranferring income from land to capital. Then, corporate income taxes and its proprietors ot benfiaciaries, then fiscal and monetary policies to promote " full employment" ( in part employment of paper pushers who then need blue collars to build the bridges into Manhatttan.) We develop higher education where most students learn nothing they will use later and potential leaders of the revolution get an income as professors.
Corn laws were rid off when the capitalists understood that, while landowners where rich like themselves, they had different long-rub interests. Capitalists agrred to share when they needed workers sppport against communism. Now, who is there to accept sharing in nthe name of theit long-run interest?
Revolution? The poors don't start revolution. They start jacquerie that are easily suppressed.
Revolution start from the rising middle class. Currently, it is not rising.

The long run scary scenario is that robots replace workers. Presumably this is scary because workers work to earn an income, if robots replace them workers no longer have an income. How is this not a sort of impossible long-run outcome? Who exactly will buy the output of the robots (presumably the output of these robots is owned by some small minority of capital owners)?

Why isn't the impossibility of this outcome a partial analogy for what is happening now and the reason inequality hurts growth. Capital earned increasing share of national income recently, it seems to me this has hit something of a limit, not just politically, but as a practical economic threshold where now in order to increase its share of the pie, the size of the pie is shrinking as demand from workers reduced incomes shrink.

OK, let's divide society into two groups, the capitalists and the working class. Imagine the capitalists own a bunch of robots that can replace most or all of the working class's economic output very cheaply. The capitalists use this to make stuff for themselves and stop hiring the working class. Bereft of income, the working class can no longer buy anything and fall into poverty.

Except, wait a second. Why don't the working class just doing the jobs they were doing before and selling the output among themselves? Maybe as soon as they do that, the capitalist will undercut them on price, and the workers will lose their jobs again. But, wait, for that to work the workers have to end up getting the stuff they would have made. If the workers all have their own self-contained non-robot economy going, there's no way the capitalists can make them worse off in aggregate by giving them cheap stuff.

Also, the workers can just make more robots for themselves, and bam, they are all capitalists.

This is where "land" (i.e. non-labor/robot resources) comes in; if production also requires non-labor/robot inputs then the capitalists could buy up all of those inputs and the workers would be unable to execute the plan above. But it's not really clear that there are such inputs (in an economically relevant way); lots of raw materials on Earth are present in effectively limitless quantities, but require increasing amounts of labor to extract (e.g. oil). Someone above mentioned asteroid mining; that's another way to transform labor into raw materials (at the very expensive end).

My question would be if capital and landowners create everything and wages continue to drop, then long term won't that create a giant Keynesian liquidity trap? If people don't buy as many goods, the capital spent and creation of robots decreases or stops. Then there is no investment needed by business or labor.

Aren't we still in a relatively global liquidity trap where the preferred savings exceeds investment?

Tiny Tim: " If the robots are building the robots that build the robots and doing so ever more cheaply. And it becomes robots all the way down. The the price of consumer goods becomes spit. Because no one has to do any labour in order for there to be a cornucopian world. Everything's made by the machines at the extreme..."

Because it's not "robots all the way down". The robots are made out of materials, run on energy and when they wear out become residual wastes -- not to mention the wastes produced during their production and the materials consumed and waste produced in the production of goods by robots. It's throughput, not robots, all the way down.

Considering that capital (robots) embodies labour and land, what happens when it embodies less labour proportionately is that it embodies proportionally more land (capital = land * labour :: land = capital/labour). You can substitute capital for land and you can substitute capital for labour but you can't substitute capital for land AND labour AT THE SAME TIME. That's the point I've being trying to raise w/regard to the Jevons paradox and the labour "fallacy of composition". The old expression was "robbing Peter to pay Paul."

Collin: No. Look at the models. The rate of interest rises over time in the first two models, which is the exact opposite of liquidity trap. In the third model the rate of interest may rise or may fall over time. It depends.

david: "Anyway. In the land model, surely the real interest rate must track the rate at which the real value of land rises? Otherwise you would borrow and buy land, or vice versa."

That is true, but you have causality reversed (in this model). We know that land rents will be rising over time as the stock of robots+human workers increases over time. And it is true that the rate of return on owning land must equal the rate of return on owning robots, and both must equal the rate of interest. But whereas the price of robots is pinned down by the linear slope of the PPF between C and I (in this simple model), there is no technology for producing more land, so the price of land is not pinned down by technology. The rate of interest determines the price of land, not vice versa.

Well... I would not say "causality reversed" inasmuch as "simultaneously determined". If we imposed desertification to steadily destroy some land, the interest rate would change anyway. But you've hashed out that debate before.

I note there is plenty of technological progress in making the marginal productivity of land larger. It's right there in the production function, since the number of robots increases.

Anyway, regardless of what pins the down the price of what, if you know the sign of the direction of change in land rents, then you know the sign of the direction of change in the interest rate.

OK, admittedly, I can barely follow any of this. But isn't the nightmare scenario the one in which

PEOPLE > L

by some significant amount? Or maybe better. Let CAP/0 be the number of people owning capital (including land) who aren't laborers, CAP/L be the number of people owning capital who also labor and L/0 be the number of people who are laborers but who own no capital. The suppose the path of the mean and total return to capital and labor goes swimmingly well, but due to the displacement of labor by technology and the concentration of the ownership of capital, we end up with.

david: "Well... I would not say "causality reversed" inasmuch as "simultaneously determined". If we imposed desertification to steadily destroy some land, the interest rate would change anyway."

Agreed.

"Anyway, regardless of what pins the down the price of what, if you know the sign of the direction of change in land rents, then you know the sign of the direction of change in the interest rate."

Disagreed. We know that Price of land = Current land rent/(interest rate - growth rate of rents). But that doesn't tell us if the interest rate is rising or falling. Price and rents might be growing at the same rate.

Dan: I admit that this post isn't easy to follow, if you aren't familiar with growth models.

I *think* I'm following you. And I *think* I would say "yes". For example, if we assumed that everyone was identical, none of my three models would be anything to worry about at all (unless we suffered horrible ennui and moral degeneracy from not really needing to work, because our income from land and robots has increased so much and is so much higher than our income from wages, which is why you need something like fox hunting to make you get up in the morning).

But in an OLG model, if some people do not inherit land or robots, and others do, it's a very different story.

Way too much to absorb and I'm not facile with this modeling language. I echo the commenter who would like to see it in (manipulable) code. Then we could play with the parameters and not have to rely on our fallible imaginations for the results.

Given that I do not understand the viability of Nick's assumption: "Let's measure wages in terms of consumption goods. Because consumption is what people care about. Robots and humans earn the same wages. Since both robots and human workers produce one unit of consumption goods per year (or per day, or per hour, or whatever) their Marginal Products and wages will be one unit of consumption goods too."

He comments in reply to Ritwick: "I can put my parameter 'a' underneath I or in front of K. It doesn't make any difference, except for vintage effects. My way of doing it handles the vintage effects better. Old robots don't get better over time. But new robots are better than old robots. So if two new robots are twice as productive as one old robot, but cost the same to produce, it's easier to redefine robots so that one new robot is really two robots, each costing half as much to produce."

Suppose as I think is reasonable (and supported by evidence) that the robots increase by some percent in efficiency (i.e. require less commodities to sustain them) each period. This is happening in industry -- energy and materials required are falling.

I can't see how this could wash out. As robots get older and are less efficient than the frontier at some point it is cheaper to replace them. Humans don't become more efficient (or at best do so much more slowly) so very quickly we get rid of them too.

Jed: computer simulations are a last resort, when we can't solve the equations. We don't need computer simulations here, because I have done the math (well, except in the last little part, but that's only because I am crap at math, and any competent economist could solve it there too).

If you want to play with the parameters, they are right there in the post!

OK, looked back at the model with this in mind. Obviously (to me) Ritwick and I are talkiing about 'W', Nick is replying in terms of 'a'. Unless there's some very cute algebra involved, we are talking past each other.

So to put the question in terms of the model: Nick, suppose the robots have a W that declines at some constant rate. What does that do to the modeled results?

Note that the "wages" of computing technology do decline at a significant rate -- my iPhone consumes a lot less energy, takes a lot less maintenance, etc. than the same capability would have a few years ago. The commodities required to sustain manufacturing processes also seem to decline but slower -- they take less energy, maintenance, space (which needs to be conditioned), etc. So I think this is a reasonable thing to explore.

I'm sure Nick will understand but just to be clear: We are *not* talking about the resources required for production but the "wages" of robots -- i.e. the commodities needed for them to operate and be maintained, just like Sraffa's circular models.

Jed: W is not a parameter. W is an endogenous variable. You can't ask that question. You have to tell me what parameter changes, then the model will tell you if W changes as a result, and what else changes too.

Jed: Aha! I get what you are asking. You aren't talking about W. You are talking about robot depreciation (the resources needed to keep robots functioning).

That's an easy(ish) question to answer.

Just change by Kdot = I equation to Kdot = I - dK where d is the rate at which robots wear out. This way of modelling it assumes that robots slowly melt, like icebergs, so you need more robots or humans to work at repairing/replacing them.

And then subtract d from the right hand-side of all my equations for the rate of interest. Or is it some multiple of g and d, like d(1-g)?

Yep. I was assuming robots have zero depreciation, or running costs. The whole of robots' wages W are paid to the owners of robots, just as the whole of a human worker's wages are paid to the owner of that human worker (himself, if it's not a slave society).

And if d falls over time, because robots require less and less maintenance, then R rises by an equivalent amount (for given K and L). No effect on W in the first model, a decrease in W in the second model, and no immediate change in W in the third model (but W will fall more quickly over time as people save and invest more now Rc is higher).

OK, I vaguely understand your replies -- thanks! But now I need more help.

Taking the last point first, owners of factories staffed by robots could undercut owners of factories staffed by humans, simply by "paying themselves" lower robot wages. And in a competitive environment they will race to the bottom. Humans can't take lower wages than subsistence so they will be priced out of the game. So all the human staffed factories go out of business, or all the humans get replaced with robots. What's wrong with that reasoning?

Regarding your first reply about W being endogenous, when laying out the model you explicitly assume robots and humans earn the same wages -- I don't think that's endogenous. Also I don't see where you derive wages from any other part of the model. So I don't know what the parameters are.

Finally the middle point: For humans, wages and "depreciation" clearly aren't independent. Humans have a minimum wage threshold for "depreciation" or their productive capacity goes to zero -- in the extreme case they die, and indeed this happens -- see Sen on famines.

You are assuming robots have no such minimum threshold. This makes the rate of dis-employment of humans extreme given the scenario in my first paragraph above -- without frictions I guess the economy jumps immediately to an equilibrium where the wages are as low as companies can survive. But even (more realistically) assuming there is such a minimum depreciation threshold for robots I believe it will fall more or less at a constant rate and so we get the same extreme result, just a bit more slowly.

PS: Thanks for making this model explicit! Having the various factors laid out with names helps a lot. Also thanks for staying on the thread and clarifying, so additional factors like depreciation can be added to the model as needed.

Jed: I don't *assume* humans and robots are paid the same wages. That is an (immediate) *implication* of the model. Look at the first equation. Humans and robots are equally productive, therefore humans and robots will be hired by profit-maximising competitive firms at wages equal to their marginal productivity (which is one unit of consumption). If human or robot W ever fell below 1, anyone could start a factory hire humans or robots, and earn a profit, increasing the demand for humans and robots until W increased to one.

Thanks again for clarifying. This is where your ability to interpret the implicit machinery of the model makes it obvious to you but unfortunately mostly opaque to me. Code would help, the machinery would have to be explicit.

What keeps the robots from being "paid" less than subsistence wages for humans -- given that their wages are really paid to the owners?

If humans are not being employed, where does demand come from? I think you say somewhere that the owners will take up any slack on consumption, but that seems like a big assumption -- though I guess it is pervasive in these models. Diminishing marginal utility (not to mention finite ability to consume) would seem to make that problematic.

I have to go do "real world" stuff for a while so won't respond further right now but I'm extremely interested in pursuing this.

Jed: OK. In my third model nothing prevents robots and humans being paid less than the human subsistence wage. Rather, it predicts it will happen.

If robot owners get satiated they will stop saving and investing. Why save and invest for a future in which you will have everything you could ever want? In which case nobody builds more robots, so technological change in building robots becomes irrelevant. And economic history grinds to a stop.

Hi,
I'm fairly knowledgeable about math but less so about economics. I'm having trouble following your first argument.

You wrote: "Let's measure wages in terms of consumption goods. Because consumption is what people care about. Robots and humans earn the same wages. Since both robots and human workers produce one unit of consumption goods per year (or per day, or per hour, or whatever) their Marginal Products and wages will be one unit of consumption goods too."

It sounds like you're just assuming that someone's wage is always equal to their marginal productivity? But for the past 40 years that hasn't been true in the US at all. And why would the robots be paid a wage at all?

Charlie: profit = P.Q - W.L - other costs where Q=F(L) (Q is output, P is price of output, W is wage, L is number of workers)

Max profit wrt L taking P and W as given. The first order condition is W/P=dF/dL (which is marginal product of labour), and in this model I have assumed P=1 (we measure prices in terms of consumption goods, not money), and dF/dL =1.

The robots get paid a wage because their owners will rent them out to the highest bidder.

Although, my intuition tells me that something is missing from the first model (although I can't tell you how I would model that). The thing is that it seems to me only logical that at one point wages should go to zero, not because labour is being substituted for, but because labour is substituting itself. I say that because you assume no population growth, thus I would assume that at one point all workers would become capital owners. It seems reasonable, since as capital owners they could make the same wage (due to the equal MP) as a worker (only that the Marginal Cost of renting out robots is probably way lower than the Marginal Cost of working - thus giving ownership the edge over labouring). Is that reasonable?

I *think* I follow you. Wages *per hour* (which is my W) would stay at 1, but total wages (W.L) would very probably go to zero eventually, because robots would become cheaper and cheaper to buy, so everyone would own one, and would choose to stop working. If I added any sensible labour supply curve and saving function to my model, that would probably happen.

Jed: Whether to buy or rent a robot depends on the rate of interest. My equilibrium condition for the rate of interest ensures people are indifferent between buying or renting a robot. Or owning a robot and renting it out, vs lending at interest. Or borrowing to buy a robot and renting it out.

It is precisely the falling price of robots, relative to constant wages of robots, that explains why the rate of interest must rise in the first two models.

NR: Who will buy the output of the robots people own? Other owners of robots, of course.

Capitalists (owners of robots) are people too! /end quote

Doesn't work. Mitt Romney's income ~=200 times mine. I own one car; he owns two, not 200. I eat three meals a day; I'm pretty sure he doesn't eat 600.

Increasing inequality in the U.S. over the last several decades has coincided with mostly unimpressive economic growth (Except for a period during the Clinton presidency when both trends reversed, IIRC).

NICK: “build a model which shows that (without land). Until then, I will say you are wrong”

(L+R)^b*K^1-b ,where R is robots and K is other forms of capital.

In your first and second scenario, we might as well assume that each individual live on a separate islands (or houses) without any need for any inputs into their production. Each persons production is, by assumption, completely unrelated to what other people do.

In your second example the wage stay the same in terms of consumption (or in terms of the amount of consumption you get the next time period from an hour of work). The present value decrease because you have the technology to transform present consumption into future consumption at a higher rate - but each persons intemporal budget set is strictly larger than in a scenario without robots – so it is hardly a nightmarish scenario (unless you account for relative wealth).

PS: I assumed that the implications of my model were obvious, but to state the obvious.

Some capital is a complement to labor (represented by K) and other forms work as a substitute (represented by R).

This mean that you might see decreasing return to capital investments (d^2Y/d(K+L)^2) and lower interest rates while a bigger share of the output goes to capital owners (and thus a smaller share to labor and lower wages).

PS 3; Furthermore, assume that the marginal benefit of consumption of goods decrease as consumption increase, but that the marginal benefit of being wealthy is constant (i.e. the benefit of feeling important).

This would mean that the interest rate kept on decreasing until only the richest (wo)man still invested – and most of those investments would not be in new capital, but would instead be him/her buying capital from people with less wealth.

PS 4: Im going to stop now but I just read your last tecnology post, and isn´t (L+R)^b*K^c with b+c<1 the obvious base for the Marxian analysis as well (given my interpretation of your statement of the theoretical problem)?

nemi: "(L+R)^b*K^1-b ,where R is robots and K is other forms of capital."

Good start, but incomplete.

If that's on the right hand side, what do you have on the left hand side of the = sign? (What replaces my "C+I/a"? It matters a lot.)

What determines the composition of investment between robots and other capital? How will that composition change over time when robots get cheaper to build?

I was toying with a model like yours for my second model, but chose what I did instead because it's so much simpler.

As far as I can remember, Marx would want either increasing returns to scale or constant returns to scale. I don't think I would want decreasing returns to scale at the level of the whole economy either. You can just replicate firms.

"Sticking land in the model gives you the equivalent of decreasing returns to scale, if you can't increase land."

Sure:
"what do you have on the left hand side of the = sign?"
Y=C+I/a
"What determines the composition of investment between robots and other capital?"
The marginal product of each component has to be equal.

"How will that composition change over time when robots get cheaper to build?"
Higher share of robots -> even lower wages.

"Sticking land in the model gives you the equivalent of decreasing returns to scale, if you can't increase land."

Yes - that is a much better model.

But I still think it is reasonable to expect decreasing return. Assume that the cost of production and sale is:

G(K,L)*F(K,L)

where F(K,L) is a constant rate of return production function while G(K,L) is a decreasing rate of return marketing function (or a function that takes a increasing part of the resourses). The more stuff the consumers consume, the less time available to evaluate each product, and the more important marketing will be.

For a given amount of saving (=Y-C) you now need to figure out the composition of investment between Rdot and Kdot. Will R be increasing over time, or K increasing over time? Or one increasing and the other decreasing?